Effects of screened electron-phonon interactions in quantum-well wires

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Effects of screened electron-phonon interactions in

quantum-well wires

To cite this article: B Tanata 1993 J. Phys.: Condens. Matter 5 2203

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J. Phys.: Condens. Matter 5 (1993) 22032212. Printed in the UK

Effects of screened electron-phonon interactions in

quantum-well wires

B ' h a t a r

Depanment of Physics, Bilkeent University, Bilkeent, 06533 ~ n l a r a , %key

Received 23 November 1992

Abstract. We study the effects of dielectric screening on scattering rates due to electron- phonon interactions in quasi-one-dimensional quantum-well wires. We consider the interaction of coniined electrons with bulk polar-optical and acoustic phonons. Acoustic phonons are examined, both in the deformation potential and the piezoelectric couplings. Screening effects are included via a temperaturedependent dielectric function q ( p )

in the random-phase approximation, by renormalizing the electron-phonon interaction. Numerical results are given for the G a b system in the quantum-size limit where a lowering of screened electron-phonon scattering rate?. is found. The effecu of screening are most significant for the electron+pticalphonon interaction.

1. Introduction

New developments in molecular beam epitaxy technology have made possible the fabrication of electronic structures based on the confinement of electrons in an essentially onedimensional (ID) semiconductor. In these synthetic materials, the electron gas is quantized in two transverse directions, so the charge carriers can only move in the longitudinal direction. Since the electron gas has a small but finite extent in the confined directions compared with the axis of free motion, such structures are also named quasi-one-dimensional ( Q ~ D ) systems. Owing to the limited number of available final states during the scattering process, the mobility of such systems is considerably enhanced, making them potentially important for high-speed device applications. Since their early prediction 111 and subsequent fabrication 12-51, there has been a lot of interest in the transport properties of Q l D systems. Scattering mechanisms due to various processes have been examined both theoretically and experimentally [7-131.

A

transport theory in I D or Q i D semiconductor structures has been developed for scattering mechanisms such as impurity [14], acoustic- phonon [7,15], and optical-phonon [S,9,15] interactions. A recent review that discusses many properties of ID semiconductor systems has also appeared [ 161.

The effects of electron-acoustic-phonon scattering in quantum-well wires were first considered by Arora [7]. Riddock and Ridley [9] and Leburton [lo] studied the electronaptical-phonon interaction

in

an infinite

quantum-well model, finding that the 1D emission rates are less than the corresponding 3D rates. The size dependence

of

scattering rates in these 1D semiconductor materials were also reported

by Leburton

[lo].

Constantinou and Ridley [17] considered

a

finite quantum-well and calculated the electron-polar-optical-phonon scattering rates within the effectivemass

(3)

approximation for a QlD cylindrical quantum-well wire. They found that the Scattering rat= were lowered compared to the infinite confining-potential approximation.

Another line of attack on the electron-phonon scattering rates for quantum-well wires has been through

Monte

Carlo simulations [lo, 18,191.

In

these calculations, optical- and acoustiophonon scattering in Q1D systems with many subbands are studied for the size and confining potential-well height dependence. Many-body calculations for self-consistent polaron scattering rates [I21 and polaronic energy [ZO] in QiD structures have also been

reported.

The scattering rates and mobilities for rectangular Q1D system due to piezoelectric phonons have also been studied [21,22]. The general conclusion of such investigations is that the piezoelectric scattering-limited mobilities in GaAs, InSb, and

InAs

are smaller than the acoustic-phonon (deformation potential coupling) limited mobility, except

for

strongly piezoelectric materials such as CdS and ZnO, where piezoelectric scattering is dominant.

Mobility measurements and calculations in semiconductor structures have always been very important in determining the various scattering mechanisms within such systems. A detailed and comparative study on phonon-limited mobility in QID semiconductors was given by Fishman 1151, in which

a

simple

model for the envelope wavefunction is employed.

We note that the transport relaxation time, T,, and the single-particle lifetime, T ~ , are in general different quantities. The relaxation time q is what enters the

mobility calculations, whereas the single-particle lifetime or the scattering time is related to the self-energy (7;'

-

ImS(IC,ck)). Only when the scattering potential

is a 6 function do we have T ,

=

T ~ , and both transport and single-particle properties

are characterized by the same quantity.

Screening effects have long been recognized as reducing the effects of the electron-phonon interaction in semiconductor structures in various dimensions. In the

usual treatment of scattering

or

relaxation-time calculations, these screening effects have been incorporated by considering the Thomas-Fermi or Debye screening lengths in the appropriate electron-phonon interaction. We notice that

for

the Thomas-Fermi screening approach to be valid, the wavevectors of interest must satisfy k Q: k,, where

IC, is the Fermi wavevector. In the case of the electron-optical-phonon interaction, wavevectors such that k

-

k , are relevant, where IC,

=

(2mwu/hZ)'12 and wo is the optical-phonon frequency. Consequently, the Thomas-Fermi static screening of the electron-phonon interaction is expected to be applicable if k , Q IC,. For GaAs heterosauctures this implies the condition n

>

1018cm-3 on the electron density (which corresponds to n

-

16cm-' for I D semiconductors). This is much higher than the electron densities typically encountered in experiment [Z]. Better approximations to screening, namely taking the full wavevector dependence of the dielectric function ~ ( q ) into account, are rarely used. Thus, motivated by the work of Constantinou and Ridley [17], we investigate the possible reduction in the scattering rates of QlD quantum-well wires due to the effects of screened electron-phonon interactions.

In this paper, our aim is to study the scattering rates due to electron-phonon interactions in Q1D structures. More specifically, we examine the electron-optical- phonon and electron-acoustic-phonon interactions. Acoustic phonons are considered both in the deformation potential and the piezoelectric couplings. We then investigate the effects of screened electron-phonon interactions on these phonon- limited scattering rates. Screening is introduced through the dielectric function

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Screened electron-phonon interacrions

in

quantum-well

wires

2205

response function for a Q1D semiconducting system has been calculated by Lee and

Spector [U]. Screening effects via a temperature-dependent ID dielectric function

~ ~ ( 4 ) on the phonon-limited mobility were considered by Fishman [IS]. We also note that electronic properties of the Q1D stmctures arising from electron-impurity

and electron-electron interactions will not be discussed here.

The rest of this paper is organized as follows. In the next section we introduce the wavefunction and energy levels for electrons confined in a cylindrical quantum wire, derive expressions

for

scattering rates due to various electron-phonon interactions, and introduce the temperature-dependent dielectric function ~ ~ ( 4 ) . In section 3 we present our numerical results on the phonon emission and absorption rates for electron-optical-phonon scattering, e1ectron;tcoustic-phonon interaction

with deformation potential coupling, and electron-acoustic-phonon interaction with piezoelectric coupling. Finally, we conclude with a brief summary.

2. Theory

We consider a model of an electron gas, quantized in two transverse directions,

so

that the charge carriers can only move in the longitudinal direction. We choose

the cross section of the system to be circular with radius R, so the quantum wire

geometry becomes cylindrical. In the quantum-size limit (QSL), the radius R of the

quantum-well wire is much smaller than the thermal de Broglie wavelength of the charge carriers, so that only the ground state (lowest) subband is populated. In the model of an infinite potential well confining the charge carriers, the normalized effective-mass wavefunction is given by

with the corresponding energy

Here

k,,

=

z , , ] / R where z?,, is the I t h root

of

the Bessel function J,(z). The ;ingle-particle scattering rates (we use the notation r

=

rs hereafter to denote the single-particle scattering time mentioned in the previous section) can be

obtained from

where the transition probability W { :

}(kz,

k;)

of an electron from a state

k,

to

k;

by emitting

or

absorbing a phonon is given by the Fermi Golden Rule

(4) 27r

h

W { : } ( k z , k i )

=

- I M { " / * S ( E ( k ~ )

-

E ( k , ) r t h w , ) .

The electron-phonon matrix element corresponding

to

the annihilation and creation of one phonon (upper and lower signs, respectively) is

(5)

where N p

is

the phonon occupation factor and the electron-phonon interaction can be written as

He-ph

=

C(q)(b,e'q.'

+

bfe-'p') (6)

in which 6,

t

and b, are the phonon creation and annihilation operators, respectively.

P

For polar optical phonons the interaction strength is

whereas for

acoustic phonons coupled through

a

deformation potential we have

and

for

acoustic phonons in the piezoelectric

coupling

In the above expressions

for

C ( q ) , 6, and c0 are the high-frequency and static

dielectric constants of the medium,

D

and I< arc the deformation potential and piezoelemic constants, v s is the sound velocity and d is the mass density.

Expressing the interaction Hamiltonian (6) in cylindrical coordinates is facilitated

by the following expression:

where we have used the following notation: q = ( q l , q,) and T

=

( p , q$, z ) . In this

work

we

consider

only

the lowest subband, thus the term involving summation over the index n in (10) does not contribute to the scattering rate. In systems where higher subbands are also occupied, the above expression can be conveniently employed.

The effects of screening have been included by renormalizing the electron-phonon interaction by the dielectric function, namely He+

-

H e + / e ( q ) . Here the dielectric function is three-dimensional, since we consider the interaction of electrons with bulk phonons. Note also that the wavevector summation in (6) is over the 3D q ValueS.

In

the approximations where a sum over q is not needed, the 1D expression for the dielectric function e ( q ) has been used [U]. We use the random- phase approximation (WA) for the static dielectric function, which is given at zero- temperature by

where a

=

(4/9rr)'I3, ' P ~

=

a / a g is the electron gas parameter, and I

=

q / 2 k p .

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Screened electron-phonon interactions in quanmm-well wires

2207

over the frequency-dependent dielectric function at zero temperature. We quote the final result for the temperature-dependent static dielectric function 1241:

where

in which EF is the Fermi energy at T

=

0, p is the chemical potential to be determined implicitly

from

the finite temperature distribution function, and k, is the Boltzmann constant. The dielectric function thus obtained does not make use of the wavefunctions of (I), therefore the geometric confinement of electrons are neglected at this level of approximation.

3. Results and discussion

We have used material parameters appropriate for GaAs to perform numerical calculations. The electron-acoustic-phonon interaction considered here within the deformation potential model is characterized by the deformation potential D

=

13.5e\! a

mass

density of the system of d

=

5.3gcnr3, and the longitudinal sound velocity in GaAs w 5

=

5.37

x

lpcms-’. For acoustic phonons in the piezoelectric coupling the dimensionless piezoelectric constant is given as

IC

=

0.052. The acoustic phonon energies here are taken to be hw,

=

hv,q.

In

the case of the electron-optical- phonon interaction, we assume the phonons are dispersionless and have energy in the bulk Lo

=

36.2meV The values of the high-frequency E , and static co dielectric constants are taken to be 10.9 and 129, respectively, and we have used the band mass of electrons, m

=

0.067me, where me is the bare electron mass. We take the electron density of the GaAs to be n

=

1.3 x l O ” ~ m - ~ , which corresponds to E, ~ i i 14meV (around zero temperature), and k,

=

0.0168;’. These numbers

indicate that only the n

=

0 level will be populated for

R

<

ISOA, making our quantum-size limit assumption valid (see also [13] for similar considerations). Also, the thermal de Broglie wavelength

A,

at T

=

300K is around 3008,

We first show

our

results

for

the electron-optical-phonon interaction. In figure 1

the inverse scattering time T-’ is depicted for two different wire radii, (a) R

=

SOA

and ( b ) R

=

100% as a function of the initial electron energy

E

in the z direction. The full curves indicate the effects of the screened electron-phonon interaction on

7-’, whereas the broken curves are obtained without screening. The upper and

lower curves show emission and absorption rates, respectively. We observe that the screening effects lower the scattering rates in general, and in the case of the

R =

SOB,

wire this reduction is about 70%. As the wire radius increases, screening effects seem slightly to increase. For instance, the reduction in the emission and absorption rates becomes about 75% for R

=

loo,& Constantinou and Ridley [IT have also found a reduction in the scattering rates (by about 25%) due to the inclusion of a finite well height for R

=

508, Note also the scales in figures l ( a ) and (b), to see the change in the scattering rate r-l as the wire radius R varies.

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10'5 o p t i c a l p h o n o n s 1014

-

?-.

----____

y1

-

.--__

_{---__}

_{---_____}

1013 10'2

---____

10'2 10" i- 0 1 2 3 4 5 0 1 2 3 4 5 E/hwo E/hoo

Figure 1. The invelse scattering time rm1 due to the elecimn-optical-phonon interaction

Cor the size quantum limit in a quantum-well wire of cixuiar cross-section, (a) with radius

R = 50A and ( b ) with radius R = looh Full and broken cuwes give r-l wilh and without screening, respectively, for emission (top awes) and absorption (boltom cuwes) p'oceues at T = 3WK

In figure 2 we present our results for the scattering rates due to the electron- acoustic-phonon interaction in the deformation potential coupling. The size dependence of 7-* (a) R

=

508, and (b) R

=

l00& is plotted as a function of the initial energy E, and we again use the polar-optical phonon energy hw, for scaling purposes. The full and broken curves indicate the scattering rates with and without dielectric screening, respectively. In contrast to the calculation for electron-ptical-phonon scattering, here we resort to the elastic approximation, namely

E(k,)

or E ( k i )

>>

fiu,, and replace the energy conservation expression

6(E(k,)

-

E(k;)

5 fiw,) by

b(E(k,)

-

E ( k : ) ) . We observe that the screening effects in the case of the electron-acoustic-phonon interaction in deformation potential coupling are not

so

pronounced as compared to the electron-optical-phonon scattering. In fact, for the R G 508, radius wire we see a 20% reduction in the

calculated scattering rate.

In

figures 3(a) and (b), we show the scattering rates due to the eleetron-acoustic- phonon interaction in the piezoelectric coupling. As in figure 2, we use TW, for scaling purposes. The full and broken curves give the scattering rates with and without dielectric screening, respectively. As in the previous cases, the effects of the screened electron-phonon interaction increases as the wire radius increases. Both for the deformation potential and piezoelectric coupled acoustic phonons we have only reported the emission rates, since at the temperature of interest here (300K) the phonon occupation factors N , and N ,

+

1 do not differ much, and the equipartition approximation is valid.

The scattering rate results we have presented here were at

T

=

300K. The relative importance

of

various electron-phonon interactions at this temperature may be assessed by the scales of figures I(a)-3(b). It has long been h o w n that the dominant phonon scattering mechanism a t low temperatures is due to acoustic phonons. At low temperatures only the emission process

takes

place, since there are not enough

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Screened eleclron-phonon interactions in

qunn

fum-weil wires 2209

(4 it,)

1 0 ~ ~ ~ ' ' ' ' ~ J " " " " ~ " ' ~ ~ " " ~ 1 0 ~ ~ ~ ~ ' " " " ' ~ " ' ' ~ " ' ~ I

' ' " I

0 1 2 3 4 5 0 1 2 3 4 5

E/hoo E/ho,

Figure 2. The inverse scattering time 7 - l due IO the electron-acoustic-phonon interaction in the deformation potential coupling for the she quantum limit in a quantum-well wire

of circular cToss-sedion. (a) with radius R = 5084 and (6) with radius R = lWA. Full and broken cufves give r - l with and wirhout rreening, respenively, for the emission

pmess at T = 300K.

10'2

acoustic phonons (PE)

----____

10'0 c 10

'

0 1 2 3 4 5 10'0 0 1 2 3 4 5 E/hwa E / b

Figure 3. The inverse scattering time r-' due to the electron-acoustic-phonon interaction in the piezoelectric coupling for the size quantum limit in a quantum-well wire of circular cross-section, ( m ) with radius R =

SO4

and ( b ) with radius R = lWA. Full and broken cufves give r-' with and wilhoul screening, respectively, for the emision p m s s at T = 3 W K .

phonons to be absorbed. It is interesting to note that the scattering rates due

to

electron-optical-phonon and electron-deformation potential coupled acoustic phonon interactions are quite comparable, within a n order of magnitude, at

T

=

300K

It has been argued [9,17] that the dimensionality effects are enhanced in 1D structures, and the singularity in the phonon emission .rate at E

=

hw, (for the electron-optical-phonon interaction) is a consequence of the ID density

of

states, which is proportional to ( E / h w , I)-'/' for emission and absorption processes,

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respectively. That there is no phonon emission below E

=

hw,

(see figures l(a) and

(b)) comes from the fact that we assume the bulk optical phonons are dispersionless, namely wq

=

w,,. S i a r density of states effects occur for electron-acoustiophonon emission rates (see figures 2(a)-3(b)), but because of the elastic approximation we employ, the onset

of

phonon emission is shifted

to

E

=

0.

To assess

the validity of the elastic approximation

(E(k,)

or E ( k : )

>

hq), we have also performed the

rate calculations for the

electron-acoustic-phonon

interaction without this assumption, finding no significant changes at the level of accuracy sought here.

In considering the effects of screening on the electron-phonon scattering rates in Q1D quantum-well wires, we have used the dielectric function ~ ( q ) in the RPA. For GaAs with carrier density n

=

1.3 x l O ” c ~ n - ~ , and assuming the effectivemass of electrons to be m

=

0.07me, we obtain rs

=

15.7. Although, strictly speaking, the RPA is valid only for vs

<

1 (the highdensity limit), we have found that the scattering rates are not altered very much by using, for example, the Hubbard approximation to the dielectric function. In figure 4 we show the temperature- dependent static dielectric function e(q) at T

=

300K (indicated by the full curve), in comparison to the zero-temperature e(q) (indicated by the broken curve) in the RPA. Evidently, the temperature dependence modifies e(q) at small q, which in turn is quite different than the non-interacting dielectric function in the same region. Examining the various forms of the electron-phonon interaction parameter C(q) (7)-

(9) and considering their contribution to the scattering time T , we expcct the effects

of screened interactions to be largest for the electron-optical-phonon scattering and smallest for the electron-acoustic-phonon scattering (deformation potential). This is clearly seen in figures l(a)-3(b). We have accounted for the temperature dependence of the dielectric function using the formulation [24] given in the previous section.

This has been crucial [15] in softening the divergence at q

=

2k, in the ID dielectric function. In our case, since we use the 3D e(q) to Screen the interaction of electrons

with bulk phonons, a temperature-dependent eT( q ) is incorporated for consistency. Indeed, we have found that the temperature dependence of the dielectric function has an effect of lowering the scattering rates. It is clear from figure 4 that the screening effects would be more significant at low temperatures than high temperatures, as noticed hy other researchers. As pointed out earlier, the model of the dielectric function we use is more appropriate to an unconfined system of electrons, since we have not evaluated eT(q) using the wavefunctions given in (1). Within the RPA dielectric function employed here, this would possibly modify the small-q region of e(q) (and consequently eT(q)) as depicted in figure 4. We believe the qualitative results reached here will remain the same

upon

the inclusion of more sophisticated models of the dielectric function.

Constantinou and Ridley [17] have found -25% reduction in the scattering rates

due to the finite depth of the confining quantum-well in Q1D semiconductor structures.

Our investigation shows a similar order-of-magnitude effect can be accounted for

by screened electron-phonon interactions. Therefore, if screened electron-phonon interactions are used with a finite quantum-well model

of

confinement, further reduction in the scattering rates is possible. The lowering in the scattering rates due to screened electron-phonon interactions, made explicit in

our

calculations, have been previously anticipated [9,25]. Although we have presented our results only for G a b , with the appropriate choice, of the material parameters different substances such

as

InAs, InSb, CdS, etc., can also be studied. Finally, we note that the relaxation time T,,

(10)

Screened elecuon-phonon mleraclions in quanhcni-well wires 2211 30 I

301~

20 n=1.3~10" cm-3 10 \ \ \ \ \

-_

0 0 0.02 0.04 0.06 0.08 0.1 n=1.3~10" cm-3 Z O I , , , , , , ,

1

10 \ \ \ \ \

-_

0 0 0.02 0.04 0.06 0.08 0.1

Figure 4. The temperaturedependent dielectric function used in the screened electron- phonon interactions. Full cuwe: q ( q ) at T = W K . broken a w e : ~ ( q ) at T = 0, both in the RPA.

interactions discussed here. The lowering

of

the scattering rates due to screened electron-phonon interactions will enhance the mobility in QID samples which will have important technological implications in terms of device applications.

4. Summary

We have studied the size effects on the screened electron-phonon interactions in Q I D quantum-well wires by calculating the scattering rates. We have considered the interaction of electrons with polar-optical and acoustic phonons. Acoustic phonons, in turn, are treated in the deformation potential and piezoelectric couplings. Screening effects are introduced by renormalizing the electron-phonon interactions with the temperature-dependent dielectric function sT(q). Effects of screened electron- phonon interactions on the single-particle scattering rates are strongest in the case of electron-optical-phonon scattering. The scattering rates due to the electron-acoustic- phonon interaction in the deformation potential coupling are found to be comparable to the electron-optical-phonon interaction.

Acknowledgments

We acknowledge the support of TUBITAK under grant number TBAG-1155. We also thank Professor Abdus Salam and the Condensed Matter Group for their hospitality at the International Centre for Theoretical Physics, 'Rieste, where part of this work has been done. Fruitful discussions with Professors M Tomak and A Erqelebi are gratefully acknowledged.

References

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Effects of screened electron-phonon interactions in quantum-well wires